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Juggling Numbers: Math Professor Develops Equation that Keeps all the Balls in the Air

ELLENSBURG, Wash. (August 7, 2012) -- The ancient art of juggling has inspired a new and growing field of mathematics. Dominic Klyve, Central Washington University math professor and ardent juggler, has developed an equation that describes all the different ways a person can juggle three balls. He, and colleagues Carsten Elsner of Universität Hannover, Institut für Mathematik, Germany, and Erik Tou of Carthage College, Wisconsin, recently published an article, "A Zeta Function of Juggling Sequences," in the Journal of Combinatorics and Number Theory.

The particular equation that he and his colleagues have formulated is based on the famous Riemann Zeta Function, possibly one of the most studied functions in math, which focuses on prime numbers. Prime numbers are special in that they are only divisible by one and themselves. They are also mysterious, in that they do not occur predictably in numerical order. The Riemann Zeta Function captures information about all the primes together in one equation, and mathematicians are still struggling with the problem of describing how this function behaves. There is a $1 million prize offered to anyone who solves it.

Based on Riemann’s work, Klyve and his colleagues developed a new zeta function. But instead of capturing information about prime numbers, it describes juggling patterns.

“This new zeta function captures information about all of the primitive juggling sequences,” said Klyve. “We call it the juggling zeta function. To be perfectly honest, we found a lot of different juggling zeta functions—there is a different one for every number of balls a person might juggle. To keep things simple, our first zeta function applies to juggling just three balls.”

Jugglers use siteswap notation to classify juggling patterns, which is based on the height to which each consecutive ball is thrown. “Height” measures the number of beats, or units of juggling time, that a ball is in the air. For example, the sequence 5;1, indicates that the first ball should be thrown to height 5, and the second to height 1, at which point the pattern repeats. Mathematicians refer to examples of siteswap notation as juggling sequences.

Instead of writing down all of the juggling sequences possible, the juggling zeta function neatly captures all the various ways three balls can be kept in the air in one elegant equation. Klyve said that the function was developed just for the “sheer fun of mathematics.” However, he admitted ruefully, it is unlikely it will ever help people juggle.

Klyve has taught mathematics at Central since 2010. He received his doctorate in mathematics from Dartmouth College in 2007. He has been juggling since 1992.